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Chapter 5: Problem 71

Percentages of public school students in fourth grade in 1996 and in eighth grade in 2000 who were at or above the proficient level in mathematics were given in the article "Mixed Progress in Math" (USA Today. August 3,2001 ) for eight western states: \begin{tabular}{lcc} State & 4th grade (1996) & 8th grade (2000) \\ \hline Arizona & 15 & 21 \\ California & 11 & 18 \\ Hawaii & 16 & 16 \\ Montana & 22 & 37 \\ New Mexico & 13 & 13 \\ Oregon & 21 & 32 \\ Utah & 23 & 26 \\ Wyoming & 19 & 25 \\ \hline \end{tabular} a. Construct a scatterplot, and comment on any interesting features. b. Find the equation of the least-squares line that summarizes the relationship between \(x=1996\) fourthgrade math proficiency percentage and \(y=2000\) eighth-grade math proficiency percentage. c. Nevada, a western state not included in the data set, had a 1996 fourth- grade math proficiency of \(14 \%\). What would you predict for Nevada's 2000 eighthgrade math proficiency percentage? How does your prediction compare to the actual eighth-grade value of 20 for Nevada?

Short Answer

Expert verified
The scatter plot, regression line and prediction are specific numerical answers that depend on the exact statistical calculations performed. To give the short answer, the step-by-step solution needs to be followed and calculations performed. The scatter plot can show a positive correlation while the accuracy of prediction can be commented based upon the difference between predicted and the actual value.

Step by step solution

01

Construct a Scatterplot

Using the provided data, construct a scatterplot with 'x' representing 1996 fourth grade math proficiency percentage, and 'y' representing 2000 eighth grade math proficiency percentage. Each state is a point on the scatterplot corresponding to its respective x and y values.
02

Comment on Scatterplot

Analyze the scatterplot's features to identify any trends or patterns. This could include identifying clusters, outliers, or whether or not there appears to be a correlation between the two proficiency percentages. The correlation could be positive, negative, or there could be no apparent correlation. Write about these observations.
03

Find the Equation of the Least-Squares Line

To summarize the relationship between the fourth and eighth-grade math proficiency percentages, compute the least-squares regression line using the given formula. This line should help to predict the eighth-grade proficiency percentage from the fourth-grade percentage.
04

Use the Regression Line to Predict

Evaluate the regression line calculated in the previous step at x = 14 to predict the 2000 eighth-grade math proficiency percentage for Nevada.
05

Compare the Prediction to the Actual

After predicting the 2000 eighth-grade math proficiency percentage for Nevada, compare the predicted value with the actual eighth-grade value of 20. Reflect on the accuracy of the prediction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot
A scatterplot is a helpful visual tool to display the relationship between two quantitative variables. In our case, it shows the relationship between the math proficiency percentages of fourth graders in 1996 and eighth graders in 2000. Each state on our list is represented as a point on this scatterplot. The x-axis represents the 1996 percentages, while the y-axis represents the 2000 percentages.

To create a scatterplot, plot each pair of values as a point. For example, Arizona corresponds to the point (15, 21). Once all states are plotted, you can start looking for patterns.

When analyzing a scatterplot, look for trends such as: - **Clusters**: Are there groups of states with similar results? - **Outliers**: Is there a state whose performance is unusual compared to others? - **Correlation**: Do the points suggest a positive, negative, or no correlation? For instance, if the points go up as you move from left to right, that indicates a positive correlation.
Least-Squares Regression Line
The least-squares regression line is a statistical method used to find the best-fitting line through a set of points in a scatterplot. This line helps us understand the relationship between two variables and make predictions about one variable based on the other.

To calculate it, we use a formula that minimizes the sum of the squares of the vertical distances from the points to the line. This method ensures the best possible fit given the data. The general equation of a line is: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.

Once you have your line, you can use it for predictions. In our case, we can predict an eighth grade proficiency percentage from a known fourth grade percentage. If you know Nevada had a fourth grade percentage of 14%, simply plug this value into your regression equation to estimate the likely eighth grade rate.
Math Proficiency Statistics
Statistics help us to make sense of educational data like math proficiency percentages. By using concepts like scatterplots and regression lines, we can visualize and quantify improvements or changes in student performance over time. For this analysis, we focus on how fourth grade results in 1996 relate to eighth grade results in 2000.

Understanding statistics means recognizing patterns in the data. For instance, if most states show an increase from fourth to eighth grade, we can infer a general improvement in math proficiency over these years. However, if a state's eighth grade proficiency is lower than or equal to its fourth grade results, it might indicate an area for concern.

Using predictive statistics, like the least-squares regression line, can inform us about expected outcomes for states not initially included in our data set. By comparing predicted proficiency values with actual ones, as we've done with Nevada, you can evaluate the reliability of your statistical tools.

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Most popular questions from this chapter

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

The article "Air Pollution and Medical Care Use by Older Americans" (Health Affairs [2002]: 207-214) gave data on a measure of pollution (in micrograms of particulate matter per cubic meter of air) and the cost of medical care per person over age 65 for six geographical regions of the United States: \begin{tabular}{lcc} Region & Pollution & Cost of Medical Care \\ \hline North & \(30.0\) & 915 \\ Upper South & \(31.8\) & 891 \\ Decp South & \(32.1\) & 968 \\ West South & \(26.8\) & 972 \\ Big Sky & \(30.4\) & 952 \\ West & \(40.0\) & 899 \\ \hline \end{tabular} a. Construct a scatterplot of the data. Describe any interesting features of the scatterplot. b. Find the equation of the least-squares line describing the relationship between \(y=\) medical cost and \(x=\) pollution. c. Is the slope of the least-squares line positive or negative? Is this consistent with your description of the relationship in Part (a)? d. Do the scatterplot and the equation of the leastsquares line support the researchers' conclusion that elderly people who live in more polluted areas have higher medical costs? Explain.

5.16 The article "California State Parks Closure List Due Soon" (The Sacramento Bee, August 30. 2009) gave the following data on \(x=\) number of visitors in fiscal year \(2007-2008\) and \(y=\) percentage of operating costs covered by park revenues for the 20 state park districts in California: \begin{tabular}{cc} & Percentage of Operating Costs Covered by Park \\ Number of Visitors & Revenues \\ \hline \(2,755,849\) & 37 \\ \(1,124,102\) & 19 \\ \(1,802,972\) & 32 \\ & \end{tabular}\begin{tabular}{rc} & Percentage of Operating Costs Covered by Park \\ Number of Visitors & Revenues \\ \hline \(1,757,386\) & 80 \\ \(1,424,375\) & 17 \\ \(1,524,503\) & 34 \\ \(1,943,208\) & 36 \\ 819,819 & 32 \\ \(1,292,942\) & 38 \\ \(3,170,290\) & 40 \\ \(3,984,129\) & 53 \\ \(1,575,668\) & 31 \\ \(1,383,898\) & 35 \\ \(14,519,240\) & 108 \\ \(3,983,963\) & 34 \\ \(14,598,446\) & 97 \\ \(4,551,144\) & 62 \\ \(10,842,868\) & 36 \\ \(1,351,210\) & 36 \\ 603,938 & 34 \\ \hline \end{tabular} a. Use a statistical software package or a graphing calculator to construct a scatterplot of the data. Describe any interesting features of the scatterplot. b. Find the equation of the least-squares regression line (use software or a graphing calculator). c. Is the slope of the least-squares line positive or negative? Is this consistent with your description in Part (a)? d. Based on the scatterplot, do you think that the correlation coefficient for this data set would be less than \(0.5\) or greater than 0.5? Explain.

The relationship between hospital patient-tonurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.

Studies have shown that people who suffer sudden cardiac arrest have a better chance of survival if a defibrillator shock is administered very soon after cardiac arrest. How is survival rate related to the time between when cardiac arrest occurs and when the defibrillator shock is delivered? This question is addressed in the paper "Improving Survival from Sudden Cardiac Arrest: The Role of Home Defibrillators" (by J. K. Stross, University of Michigan, February 2002 ; available at www.heartstarthome.com). The accompanying data give \(y=\) survival rate (percent) and \(x=\) mean call-toshock time (minutes) for a cardiac rehabilitation center (in which cardiac arrests occurred while victims were hospitalized and so the call-to-shock time tended to be short) and for four communities of different sizes: \(\begin{array}{llllll}\text { Mean call-to-shock time, } x & 2 & 6 & 7 & 9 & 12\end{array}\) Survival rate, \(y\) \(\begin{array}{lllll}90 & 45 & 30 & 5 & 2\end{array}\) a. Construct a scatterplot for these data. How would you describe the relationship between mean call-toshock time and survival rate? b. Find the equation of the least-squares line. c. Use the least-squares line to predict survival rate for a community with a mean call-to-shock time of 10 minutes.
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